International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017

A Study on Methods of Contour Integration Of Complex Analysis

Abdulsattar Abdullah Hamad, Master of mathematics Department of mathematics College of Science, Acharya nagarjuna University Email id: [email protected] ABSTRACT: -Application of the Cauchy integral formula

Complex analysis is considered as a -Application of the residue theorem. powerful tool in solving problems in One method can be used, or a combination mathematics, physics, and engineering. In of these methods or various limiting the mathematical field of complex analysis, processes, for the purpose of finding these contour integration is a method of evaluating integrals or sums. certain integrals along paths in the complex INTRODUCTION: plane. Contour integration is closely related Cauchy is considered as a principal founder to the calculus of residues, a method of of complex function theory. However, the complex analysis. One use for contour brilliant Swiss mathematician Euler (1707- integrals is the evaluation of integrals along 1783) also took an important role in the field the real line that are not readily found by of complex analysis. The symbol 𝑖 for √−1 using only real variable methods. Contour was first used by Euler, who also introduced integration methods include: 휋 as the ratio of the length of a

circumference of a circle to its diameter and -Direct integration of a complex –valued 푒 as a base for the natural logarithms, function along a curve in the complex plane

(a contour) respectively. He also developed one of the

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International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 most useful formulas in mathematics. That laid the foundation from which another is Euler’s formula 푒푖휃 = cos 휃 + sin 휃 , mathematical giant, Cauchy, developed the which can be derived from De Moivre’s full scale theory of complex analysis and its theorem [1]. applications.

Letting 휃 = 휋 in Euler’s formula, we obtain Complex analysis is essentially the study of the equation 푒푖휋 + 1 = 0. The equation, complex numbers, their derivatives, and

푒푖휋 + 1 = 0 , contains what some integrals with many other properties. On the mathematicians believe as the most other hand, number theory is concerned with significant numbers in all of mathematics, the study of the properties of the natural i.e. . 푒, 휋, 𝑖 And the two integers o and 1. It numbers. These two fields of study seen to

휋 푖 − be unrelated to each other. However, in also gave the remarkable result 𝑖 = 푒 2 , number theory, it is sometimes impossible to which means that an imaginary power of an prove or solve some problems without the imaginary number can be a real number , use complex analysis. The same is true in and he showed that the system of complex many problems in applied mathematics, numbers is closed under the elementary physics and engineering. In a calculus transcendental operations. Moreover, Euler course, the exact value of some real –valued was the first person to have used complex definite integrals cannot be computed by analysis methods for trying to prove finding its anti-derivative because some Fermat’s last theorem, and he actually anti-derivative is impossible to find in terms proved the impossibility of integer solutions of elementary functions. In these cases, the of 푥 3 + 푦3 = 푧3. Euler’s contribution to integrals are only calculated numerically. complex analysis is significant. His work

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Amazingly, complex analysis enables us to The French mathematician Augustin-Louis evaluate many of these integrals. One Cauchy (1789-1857) brought great important application of complex analysis is contributions to mathematics, providing to find the values of real definite integrals by foundations for mathematical analysis, calculating contour integrals in the complex establishing the limit concept and general plane [2]. theory of convergence, and defining the

definite integral as the limit of a Sum. A real definite integral is an integral

푏 푓(푥)푑푥 , where 푓(푥) is a real-valued Cauchy also devised the first systematic ∫푎 function with real numbers a and b. Here, a theory of complex numbers. Especially, his best-known work in complex function general definite integral is taken in the

theory provides the Cauchy integral theorem complex plane, and we have 푓(푧)푑푧 ∫푐 as a powerful tool in analysis. Moreover, his where c is a contour and z is a complex discovery of the calculus of resides is variable. Contour integration provides a extremely valuable and its applications are method for evaluating integrals by marvelous because it can be applied to the investigating singularities of the function in evaluation of definite integrals, the domains of the complex plane. The theory of summation of series , solving ordinary and the functions of a real variable had been partial differential equations , difference and developed by Lagrange, but the theory of algebraic equations, to the theory of functions of a complex variable had been symmetric functions , and moreover to achieved by the efforts of Cauchy. mathematical physics.

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In the resume, Cauchy gave his definition of CONTOUR INTEGRAL the derivative as a limit and defined the We turn now to integrals of complex-valued definite integral as the limit of a sum. As a functions 푓 of the complex variable 푧. Such consequence, Weirstrass and Riemann an integral is defined in terms of the values extended his work on complex function 푓(푧) along a given contour 퐶 , extending theory, and also Riemann and Lebesgue from a point 푧 = 푧1 to a point 푧 = 푧2 in improved on his definition of the integral. In the complex plane. It is, therefore, a line his book, he also retained the mean-value integral; and its value depends, in general, theorem much as Lagrange had derived it. on the contour 퐶 as well as on the function Unlike Lagrange, Cauchy used tools from 푓. It is written integral calculus to obtain Taylor’s theorem

푧2 [3]. ∫ 푓(푧)푑푧 표푟 ∫ 푓(푧) 푑푧, 푐 푧1 Cauchy also did more with convergence The latter notation often being used when properties for Taylor’s series. In the ‘’ cours the value of the integral is independent of d’analyse ‘’, he had formulated the Cauchy the choice of the contour taken between two criterion independently of Bolzano early in fixed end points. While the integral may be 1817. In his writings, Cauchy proved and defined directly as the limit of a sum, we often utilized the ratio, root, and integral choose to define it in terms of a definite tests, thereby establishing the first general integral . Suppose that the equation theory of convergence. Among those great works, the importance of Cauchy’s residue 푧 = 푧(푡) (푎 ≤ 푡 ≤ 푏) (1.2.5.1) theory cannot be underestimated.

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Represents a contour 퐶, extending from a neighborhood of 푧0 . It follows that if 푓 is point 푧1 = 푧(푎) to a point 푧2 = 푧(푏). We analytic at a point 푧0, it must be analytic at assume that 푓[푧(푡)] is piecewise continuous each point in some neighborhood of 푧0 . A on the interval 푎 ≤ 푡 ≤ 푏 and refer to the function f is analytic in an open set if it has a function 푓(푧)as being piecewise continuous derivative everywhere in that set. If we on 퐶 [4]. We then define the line integral, or should speak of a function 푓 that is analytic contour integral, off along 퐶 in terms of the in a set 푆 which is not open, it is to be parameter 푡: understood that f is analytic in an open set

containing 푆. 푏 ∫ 푓(푧) 푑푧 = ∫ 푓[푧(푡)] 푧′(푡)푑푡. (1.2.5.2) 푐 푎 Note that the function 푓(푧) = 1 is analytic at 푧

Note that since 퐶 is a contour, 푧′(푡) is also each nonzero point in the finite plane. But piecewise continuous on 푎 ≤ 푡 ≤ 푏; and the function 푓(푧) = | 푧|2 is not analytic at so the existence of integral (1.2.5.2) is any point since its derivativeexists only at ensured. The value of a contour integral is 푧 = 0 and not throughout any invariant under a change in the neighborhood. An entire function is a representation of its contour. function that is analytic at each point in the

ANALYTIC FUNCTIONS entire finite plane. Since the derivative of a polynomial exists everywhere, it follows We are now ready to introduce the concept that every polynomial is an entire function. of an analytic function. A function 푓 of the If a function 푓 fails to be analytic at a point complex variable 푧 is analytic at a point 푧0 if 푧0 but is analytic at some point in every it has a derivative at each point in some neighborhood of 푧0 , then 푧0 is called a

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International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 singular point, or singularity, of 푓. The point ∞ 푛 푓(푧) = ∑ 푎푛 (푧 − 푧0) (|푧 − 푧0| 푧 = 0 is evidently a singular point of the 푛=0

1 function 푓(푧) = . The function 푓(푧) = < 푅0 ),(1.2.7.1) 푧

| 푧|2 , on the other hand, has no singular where points since it is nowhereanalytic. 푓(푛)(푧 ) 푎 = 0 (푛 = 0,1,2,. . . ). (1.2.7.2) 푛 푛! SERIES (LAURENT SERIES)

That is, series (1.2.7.1) converges to We turn now to Taylor’s theorem, which is 푓(푧)when 푧 lies in the stated open disk. one of the most important results of this section. This is the expansion of 푓(푧)into a Taylor

series about the point 푧0 . It is the familiar Theorem. Taylor series from calculus, adapted to

Supposethatafunction 푓 functions of a complex variable. With the isanalyticthroughoutadisk |푧 − 푧0| < 푅0 , agreement that 푓(0)(푧0) = 푓(푧0) and 0! = centered at and with radius . Then 푧0 푅0 1, series (1.2.7.1) can, of course, be written 푓(푧)has the power series representation

푓′(푧 ) 푓′′(푧 ) 푓(푧) = 푓(푧 ) + 0 (푧 − 푧 ) + 0 (푧 − 푧 )2 +· (|푧 − 푧 ) < 푅 ). (1.2.7.3) 0 1! 0 2! 0 0 0

Any function which is analytic at a point 푧0 point ; and 휀 may serve as the value of 푅0in must have a Taylor series about 푧0. For, if 푓 the statement of Taylor’s theorem. Also, if 푓 is analytic at 푧0 , it is analytic throughout is entire,푅0 can be chosen arbitrarily large; some neighborhood |푧 − 푧0| < 휀 of that and the condition of validity becomes |푧 −

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∞ (푛) 푧0| < ∞. The series then converges to 푓(푧) 푓 (0) 푓(푧) = ∑( ) 푧 (|푧| 푛! 푛 at each point 푧 in the finite plane. When it is 푛=0 known that f is analytic everywhere inside a < 푅0 ).(1.7.2.4) circle centered at 푧 , convergence of its 0 LAURENT SERIES

Taylor series about 푧0 to 푓(푧) for each point If a function 푓 fails to be analytic at a 푧 within that circle is ensured; no test for the

point푧0, one cannot apply Taylor’s theorem convergence of the series is even required. at that point. It is often possible, however, to In fact, according to Taylor’s theorem, the find a series representation for 푓(푧) series converges to 푓(푧) within the circle involving both positive and negative powers about 푧0 whose radius is the distance from

of 푧 − 푧0 . We now present the theory of 푧0 to the nearest point 푧1at which f fails to such representations, and we begin with be analytic. we shall find that this is actually Laurent’s theorem. the largest circle centered at 푧0 such that the series converges to 푓(푧) for allz interior to Theorem it. In the following section [5], we shall first Suppose that a function 푓 is analytic throughout an annular domain prove Taylor’s theorem when 푧0 = 0, in which case f is assumed to be analytic 푅1 < |푧 − 푧0| < 푅2 , centered at 푧0 , and let 퐶 denote any positively oriented simple throughout a disk |푧| < 푅0 and series closed contour around 푧0 and lying in that (1.7.2.1) becomes a Maclaurin series: domain . Then, at each point in the domain, 푓(푧)has the series representation

∞ ∞ 푛 푛 푓(푧) = ∑ 푎푛 (푧 − 푧0) + ∑ 푏푛 (푧 − 푧0) (푅1 < |푧 − 푧0| < 푅2), 푛=0 푛=0

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Where

1 푛 +1 푎푛 = ∫ (푓(푧)푑푧/(푧 − 푧0) ) (푛 = 0,1,2, . . . ) 푎푛푑 2휋𝑖 푐

1 −푛+1 푏푛 = ∫ ( 푓(푧)푑푧/(푧 − 푧0) ) (푛 = 1,2, . . . ). 2휋𝑖 푐

CAUCHY DISTRIBUTION

The integral

(which arises in probability theory as a scalar multiple of the characteristic function of the Cauchy distribution) resists Since eitz is an entire function (having the techniques of elementary calculus. We no singularities at any point in the complex will evaluate it by expressing it as a limit of plane), this function has singularities only contour integrals along the contour C that where the denominator z2 + 1 is zero. goes along the real line from −a to a and Since z2 + 1 = (z + i)(z − i), that happens then counterclockwise along a semicircle only where z = i or z = −i. Only one of those centered at 0 from a to −a. Take a to be points is in the region bounded by this greater than 1, so that the imaginaryunit i is contour [6]. The residue off(z) at z = i is enclosed within the curve. The contour integral is

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According to the residue theorem, then, we have

The contour C may be split into a "straight" part and a curved arc, so that

and thus

It can be shown that if t > 0 then

Therefore if t > 0 then

A similar argument with an arc that winds around −i rather than i shows that if t < 0 then

and finally we have this:

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(If t = 0 then the integral yields immediately to real-valued calculus methods and its value is π.)

If we call the arc of the semicircle Arc, we need to show that the integral over Arc tends to zero as a → ∞ — using the estimation lemma

where M is an upper bound on |f(z)| along the Arc and L the length of Arc. Now,

So

APPLICATION TO CALCULATE THE The residue theorem can also calculate some SUMS sums endless. Is a function g having for each residue integer n equal to n-th general term In 1831, Cauchy announced the theorem that of an infinite sum S and a set E of residues an analytic function of a complex variable corresponding to other points. Suppose that 푤 = 푓(푧) can be expanded about a point the integral of this function along a lace 훾 푧 = 푧 in a power series that is convergent 0 correctable infinitely large is zero [7]. There for all values of 푧 writin a circle having 푧 0 is then the residue theorem: as center with a radius. 푟 > 0 From this time on, the use of infinite series became an essential part of the theory of functions of both real and complex variables.

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Therefore, one can express the infinite sum by another sum (usually finite) residue:

And

Statements below provide more general examples of cases where this method is applicable: the sum of the "first type Proof

We have

the sum of the "second type

By using the integral test of convergence we First type observe that this sum converge .

Let the evaluation of the following sum: We use the same argument to prove that the sum

with 푓 having a set 퐸 of singularity . Suppose that the following conditions are Converge: respected : As we avoid the set 퐸 of singularity of 푓 in There exist 푀,푅 > 0 and 훼 > 1 such that the sum , we have only

|푓(푧)| ≤ 푀 for all 푧 complex of modulus |푧|훼 greater than or equal to 푅.

Then , we have : (finite sum of bonded terms ) and finally:

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Suppose that 푔(푧) = 푓(푧)휑(푧)

It must then the function 휑 has a simple pole

We have to find a function 푔 which its of residue 1 at each integer . residues are {푓(푛), 푛 ∈ ℤ}.

A function having this property is given by:

Then ,sin (휋푧) has a simple zero for each integer 푧 and

Where we have used the residue formula for the fraction having simple zero to the denominator .

Take for the contour the circle center at the origin and of radius 푅 = 푁 + 0.5 with 푁 ∈ ℕ and the increment of one half proving that we avoid the pole located on ±푁.

At the limit [8], the residue theorem give:

It remains us now to prove that that limit is null to get the result we want.

By using the estimation lemma, we have:

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The modulus of the function cot is bounded by a some constant 퐾 > 0 on the contour because we avoid the integers of the real axes of the choice of contour , the right member of the inequality below is majored by

Where we have used the reason that > 1 . as the limit is as well zero , the result is proved.

Second type

Let the calculus of the following sum:

With 푓 having a set 퐸 of isolated singularity .

Suppose that 푓 satisfy at the same condition that for the sum of the first type :

There exist

Such that

For all complex 푧 of the modulus greater than or equal to 푅.

Then, the sum converge absolutely and we have:

Proof

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The proof of identic to the type , it enough to prove that the function

Has as residue

.

We have

With a simple pole at each integer point.

The residue of the fraction having a simple zero at the denominator is given by:

Which complete the proof.

SUMMATION OF SERIES USING COMPLEX VARIABLES

Another way to sum infinite series involves the use of two special complex functions, namelywhere 푓(푧) is any function with a finite number of poles at

푧1 ,푧2 ,. . 푧푁within the complex plane and cot(B z) and csc(Bz) have the interesting property that they have simple poles at all the intergers 푛 = −4, . . . ,0, . . . +4 along the real z axis. One knows from Cauchy’s residue theorem that the closed line contour enclosing all the poles of functions 퐹(푧) and 퐺(푧) equals 2 휋𝑖 times the sum of the residues. If we now demand that both F(z) and

G(z) vanish on a rectangular contour enclosing all the poles , where again the 푧푛 refers to the location of the N poles of 푓(푧). In deriving these results we have made use of the well known result that the residue for first order poles of 푔(푧)/ℎ(푧) at the zeros of ℎ(푧) is simply 푔(푧푛 )/

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ℎ’(푧푛 ). Let’s demonstrate this summation approach for several classical examples. Look first at the function 푓(푘) = 1 where 푓(푧) has poles at 푧 = 𝑖푎 and [9] 푎2+푘2 1

푧2 = −𝑖푎 . Plugging into the first residue formula above, we have

+∞ 1 휋cot (𝑖푎휋) 휋 cot(−𝑖푎휋) 휋 ∑ = − − = coth (휋푎) 푘2 + 푎2 2𝑖푎 −2𝑖푎 푎 푘=−∞

or, noting the even symmetry of the quotient 푐표푡(휋푎)/푎, that

+∞ 1 1 휋 ∑ = − + coth (휋푎) 푘2 + 푎2 2푎2 2푎 푘=1

If one takes the limit as a approaches zero(done by using the series expansions about 푎 = 0 for cosine and sine plus application of the geometric series) the famous result of Euler that the sum of the reciprocal of the square of all positive integers is equal to 휋 2 /6is obtained.

As the next example look at 푓(푘) = (1/(푘^2푚).

Here we have just a single 2푚 th order pole at 푧 = 0 and one finds

4 6 8 ∑∞ 1 = 휋 , ∑∞ 1 = 휋 , ∑∞ 1 = 휋 푘=1 푘4 90 푘=1 푘6 945 푘=1 푘8 9450 for푚 = 2, 3 and 4, respectively.

Next we look at a series with alternating signs. For the following case we get

∞ (−1)푘 휋csc (휋푧)) 휋 ∑ = −푅푒푠 [ , 푧 = ±𝑖푎] = (푘2 + 푎2) 푧2 + 푎2 asinh (휋푎) 푘=−∞ which allows one to state that

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1 1 1 (−1)푘 휋 − + + ⋯ . = 5 10 17 1 + 푘2 2sinh (휋) when a=1.

All of the above examples have involved Evaluate the following integral by using even functions 푓(푘) . One now asks what residue method about odd functions such as (푘) = 1/푘3 ? Although the above residue formulas do not apply to odd functions, a modification is possible as we now show. Consider the Solution function 퐻(푧) = 휋 푠푒푐(휋푧)푓(푧) , where The function has 2 simple poles 푝1,2 = ±𝑖푎 . 푓(푧) is now an odd function, and then make One of these 2 pole is inside the upper plan, a closed line contour integration of 퐻(푧) about the rectangular contour with corners at we have then 퐼=2𝑖 휋 푅푒푠 ( ,푎) (푁 + 1/2)(1 + 𝑖), (푁 + 1/2)(−1 + 𝑖), With

(푁 + 1/2)(−1 − 𝑖), and (푁 + 1/2)(1 − 𝑖).

This leads towhere the left integral vanishes as N goes to infinity, the residue for 퐻(0) becomes 휋 3 /2 when f(푧) = 1/푧3 , and the Therefore residues 퐻(푛 + 1/2) become 1/ 3 [푛 + 1) 푠𝑖푛(휋(푛 + 1/2))] .We thus have 2 the interesting result that-[10] CONCLUSION: ∞ 휋 3 (−1)푛 = ∑ In this project we will apply the powerful 32 (2푛 + 1)3 푛=0 1 1 1 1 technique of Contour integral in the complex = − + − + ⋯ 13 33 53 73 plane to evaluate some improper integrals.

These integrals are very difficult to tackle

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International Journal of Research p-I SSN: 2348-6848 e-I SSN: 2348-795X Available at Volume 04 Issue 03 https://edupediapublications.org/journals Ma rch 2017 with the regular calculus techniques of real integral. Japan Journal of Industrial and variables. We are going to use the Applied Mathematics, 27(1), 73-90. integration along a branch cut, and the 5. Schempp, W. (1982). Complex contour residue theorem, plus the proper choice of integral representation of cardinal spline contours, to solve interesting integrals. functions (Vol. 7). American

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